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  "body": "\n## Example\n\nThe compact Lie group $E_8$ is the compact, connected and simply connected [[simple Lie group|AM7YJP8AR6]] whose Lie algebra has the root system of type $E_8$.\nThe root system $E_8$ lives in $\\R^8$ and consists of the $240$ vectors of the following two kinds:\n\n1. the $112$ vectors with two entries $\\pm 1$ and six entries $0$;\n2. the $128$ vectors $(\\pm \\tfrac12, \\dots, \\pm \\tfrac12)$ with an even number of minus signs.\n\nEvery root has squared length $2$.\nCheck of the count: there are $\\binom{8}{2} \\cdot 4 = 112$ vectors of the first kind and $2^8 / 2 = 128$ of the second kind.\n\nThe dimension of $E_8$ is the rank plus the number of roots, $8 + 240 = 248$.\n\n## Remarks\n\n$E_8$ is the largest of the five exceptional compact simple Lie groups $G_2, F_4, E_6, E_7, E_8$, see [[G2|YX5FFSCR8P]] for the smallest.\nIts Dynkin diagram is a tree with $8$ nodes and branch lengths $1$, $2$ and $4$ from the branch point.\nThe compact group $E_8$ has trivial centre, so it is at the same time simply connected and adjoint.\nIts smallest nontrivial representation is the adjoint representation, of dimension $248$.\nThe $240$ roots are the shortest nonzero vectors of the [[E8 lattice|2CXH65EKY7]], which is spanned by them.\nThe Weyl group of $E_8$ has order $696\\,729\\,600$.\nThe name $E_8$ also denotes the complex simple Lie group of this type and its real forms.\n\n## Explanations\n\n### L1 human\n\n$E_8$ is the largest and most intricate of the exceptional Lie groups, the ones that do not belong to the infinite families of matrix groups.\nIt has dimension $248$, yet it is built from a very symmetric arrangement of $240$ vectors in eight-dimensional space.\nThe lattice that these vectors span gives the densest packing of equal balls in eight dimensions.\nThe group appears in many places, from the classification of lattices to string theory.\n",
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